Products of finite connected subgroups

TL;DR

This paper generalizes Thompson's theorem by studying finite groups formed as products of subgroups with specific properties, extending results to classes like metanilpotent groups and groups with nilpotent derived subgroups.

Contribution

It introduces the concept of $ ext{L}$-connected products and extends previous results to new classes of groups, providing characterizations and local subgroup descriptions.

Findings

$[A,B]$ is soluble when $ ext{L}= ext{S}$ for soluble groups

Characterization of connected products for metanilpotent groups

Local subgroup descriptions in finite groups

Abstract

For a non-empty class of groups $\cal L$, a finite group $G = AB$ is said to be an $\cal L$-connected product of the subgroups $A$ and $B$ if $\langle a, b\rangle \in \cal L$ for all $a \in A$ and $b \in B$. In a previous paper, we prove that for such a product, when $\cal L = \cal S$ is the class of finite soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In particular, we characterize connected products for relevant classes of groups; among others the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Also we give local descriptions of relevant subgroups of finite groups.